Problem 4, 2024

Let \(k \geq 2\) be a natural number. Margita has written on the board the first \(2k - 1\) natural numbers \(1, 2, \ldots, 2k - 1\). In one move she may erase any two numbers from the board and write on it the sum and the product of the erased numbers.

Let \(n \geq 2k + 1\) be a given odd natural number. Can Margita play her moves so that at the end exactly \(k\) occurrences of the number \(n\) remain on the board?